Thread: Balls and Buckets

1. Balls and Buckets

There are 5 buckets labeled A, B, C, D and E and 5 balls labeled a, b, c, d and e. Given that each bucket contains 1 and only 1 ball, what is the probability that ball "a" ends up in buckets "A" if none of the other buckets contain balls with a matching letter?

2. Re: Balls and Buckets

<img src=/S/confused.gif border=0 alt=confused width=15 height=20>Is it <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">relevant what happens in the other buckets</span hide> <img src=/S/question.gif border=0 alt=question width=15 height=15> The probability that bucket A contains ball a is <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"> 1 in 5 </span hide> or <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">20</span hide>% That is depending upon how the balls ended up in the bucket - For example if I asked John Grey to put ball a in bucket A the probability would <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">increase to 100</span hide>% n'est pas?

3. Re: Balls and Buckets

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9 / 53</span hide>

4. Re: Balls and Buckets

I get the same as Hans <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9/53</span hide>

Steve

5. Re: Balls and Buckets

<P ID="edit" class=small>(Edited by AlanMiller on 02-Dec-04 08:55. )</P>I make it <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1/4</span hide>, but I haven't woken up yet. <img src=/S/doze.gif border=0 alt=doze width=15 height=15> <img src=/S/yawn.gif border=0 alt=yawn width=15 height=15>

Now, having shaken some cranial cells, I think the wording needs some qualification. Is the question asking:

o What is the probability that ball "a" ends up in buckets "A" given that none of the other buckets contain balls with a matching letter?
or
o What is the probability that ball "a" ends up in buckets "A" and none of the other buckets contain balls with a matching letter?

Alan

6. Re: Balls and Buckets

I make it <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.075</span hide>

7. Re: Balls and Buckets

I fail to see the difference in the 2 statements?

I determined (as well as Hans, who posted before me) that there were <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">53</span hide> possibilities where none of B-E had the matching ball. In <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> of them A had its matching ball.

Steve

8. Re: Balls and Buckets

The difference being conditional probability, and the two represent different numbers of possible configurations. One asks "Given that none of the buckets B,C,D,E contain their corresponding balls, what is the probability that A contains a?" The other asks "Of all the possible configurations, what is the probability that A contains a, and none of the buckets B,C,D,E contain their corresponding balls?" They are two quite different event spaces, the first being a subset of the second. It's the "if" qualifier in the question that needs further qualification.

Your solution is for the first situation, which is the way I'd now interpret the question (with the "if" indicating "given that").

Alan

9. Re: Balls and Buckets

I also read the if as "given that".

10. Re: Balls and Buckets

If I relook at your descriptions: are the 2 possibilities:

So your "first" statement is the <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> possibilities of A having a and none of the others having a match out of the "SUBSET" of <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">53</span hide> where none of the others was matching.

And your "second" statement is the <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> possibilities of A having a and none of the others having a match out of the "TOTAL" set of <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">120</span hide> possibilities.

Am I interpreting that correct?

As Hans mentioned, I saw the first based on "if none" as "given".

So much "subtlety" and "interpretation" in language...
Steve

11. Re: Balls and Buckets

Sorry, I cannot help you with the clarification. I lifted this from a multiple choice exam that some students were discussing. The model solution was 20% which everyone disagreed with but they hadn't settled on an answer. I worked out the same answer as Hans / Steve after reading the 'if' as 'given that'. I don't think it a is valid interpretation reading 'if' as 'and'.

12. Re: Balls and Buckets

The solution is "close to" 20%.

If none of the buckets holds its "match" there are 44 possibilities. Each ball is in a bucket 11 times. If you "try to extend" the logic and say therefore the should be 11 cases with "a" being in A to make 11/(44+11) = 11/55 = 1/5 = 20%

The reality is that there are only <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> cases where A has a and the others have none of the match to give <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9/(44+9) = 9/53 = 17.0%</span hide>

Steve

13. Re: Balls and Buckets

Well done Steve - I think you have solved it both ways (correct answer and faulty logic to show how they got to 20%).

14. Re: Balls and Buckets

And I agree with absolutely everything that Alan says! Oh yes. <img src=/S/confused3.gif border=0 alt=confused3 width=45 height=45>

15. Re: Balls and Buckets

Hey <img src=/S/exclamation.gif border=0 alt=exclamation width=15 height=15> Didn't I get there first <img src=/S/question.gif border=0 alt=question width=15 height=15> <img src=/S/blowup.gif border=0 alt=blowup width=60 height=60>

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